Antarctic Whale Project: Single Species
Ben Weinstein
2016-10-19 10:13:10
0.1 By Month
0.2 CCAMLR Units
0.3 Distance
0.4 Time
0.5 Velocity
0.6 Angles
1 Correlated random walk
Process Model
\[ d_{t} \sim T*d_{t-1} + Normal(0,\Sigma)\] \[ x_t = x_{t-1} + d_{t} \]
1.1 Parameters
For each individual:
\[\theta = \text{Mean turning angle}\] \[\gamma = \text{Move persistence} \]
For both behaviors process variance is: \[ \sigma_{latitude} = 0.1\] \[ \sigma_{longitude} = 0.1\]
1.2 Behavioral States
\[ \text{For each individual i}\] \[ Behavior_1 = \text{traveling}\] \[ Behavior_2 = \text{foraging}\]
\[ \alpha_{i,1,1} = \text{Probability of remaining traveling when traveling}\] \[\alpha_{i,2,1} = \text{Probability of switching from Foraging to traveling}\]
\[\begin{matrix} \alpha_{i,1,1} & 1-\alpha_{i,1,1} \\ \alpha_{i,2,1} & 1-\alpha_{i,2,1} \\ \end{matrix}\]
With the probability of switching states:
\[logit(\phi_{traveling}) = \alpha_{Behavior_{t-1}}\]
\[\phi_{foraging} = 1 - \phi_{traveling} \]
1.3 Continious tracks
The transmitter will often go dark for 10 to 12 hours, due to weather, right in the middle of an otherwise good track. The model requires regular intervals to estimate the turning angles and temporal autocorrelation. As a track hits one of these walls, call it the end of a track, and begin a new track once the weather improves. We can remove any micro-tracks that are less than three days. Specify a duration, calculate the number of tracks and the number of removed points. Iteratively.
How did the filter change the extent of tracks?
Look at the observations were defined into tracks.
sink(“Bayesian/Multi_RW.jags”) cat(" model{
#Constants
pi <- 3.141592653589
##argos observation error##
argos_prec[1:2,1:2] <- inverse(argos_sigma*argos_cov[,])
#Constructing the covariance matrix
argos_cov[1,1] <- 1
argos_cov[1,2] <- sqrt(argos_alpha) * rho
argos_cov[2,1] <- sqrt(argos_alpha) * rho
argos_cov[2,2] <- argos_alpha
for(i in 1:ind){
for(g in 1:tracks[i]){
## Priors for first true location
#for lat long
y[i,g,1,1:2] ~ dmnorm(argos[i,g,1,1,1:2],argos_prec)
#First movement - random walk.
y[i,g,2,1:2] ~ dmnorm(y[i,g,1,1:2],iSigma)
###First Behavioral State###
state[i,g,1] ~ dcat(lambda[]) ## assign state for first obs
#Process Model for movement
for(t in 2:(steps[i,g]-1)){
#Behavioral State at time T
phi[i,g,t,1] <- alpha_mu[state[i,g,t-1]]
phi[i,g,t,2] <- 1-phi[i,g,t,1]
state[i,g,t] ~ dcat(phi[i,g,t,])
#Turning covariate
#Transition Matrix for turning angles
T[i,g,t,1,1] <- cos(theta[state[i,g,t]])
T[i,g,t,1,2] <- (-sin(theta[state[i,g,t]]))
T[i,g,t,2,1] <- sin(theta[state[i,g,t]])
T[i,g,t,2,2] <- cos(theta[state[i,g,t]])
#Correlation in movement change
d[i,g,t,1:2] <- y[i,g,t,] + gamma[state[i,g,t]] * T[i,g,t,,] %*% (y[i,g,t,1:2] - y[i,g,t-1,1:2])
#Gaussian Displacement
y[i,g,t+1,1:2] ~ dmnorm(d[i,g,t,1:2],iSigma)
}
#Final behavior state
phi[i,g,steps[i,g],1] <- alpha_mu[state[i,g,steps[i,g]-1]]
phi[i,g,steps[i,g],2] <- 1-phi[i,g,steps[i,g],1]
state[i,g,steps[i,g]] ~ dcat(phi[i,g,steps[i,g],])
## Measurement equation - irregular observations
# loops over regular time intervals (t)
for(t in 2:steps[i,g]){
# loops over observed locations within interval t
for(u in 1:idx[i,g,t]){
zhat[i,g,t,u,1:2] <- (1-j[i,g,t,u]) * y[i,g,t-1,1:2] + j[i,g,t,u] * y[i,g,t,1:2]
#for each lat and long
#argos error
argos[i,g,t,u,1:2] ~ dmnorm(zhat[i,g,t,u,1:2],argos_prec)
}
}
}
}
###Priors###
#Process Variance
iSigma ~ dwish(R,2)
Sigma <- inverse(iSigma)
##Mean Angle
tmp[1] ~ dbeta(20, 20)
tmp[2] ~ dbeta(10, 10)
# prior for theta in 'traveling state'
theta[1] <- (2 * tmp[1] - 1) * pi
# prior for theta in 'foraging state'
theta[2] <- (tmp[2] * pi * 2)
##Move persistance
# prior for gamma (autocorrelation parameter)
#from jonsen 2016
##Behavioral States
gamma[1] ~ dbeta(4,2) ## gamma for state 1
dev ~ dbeta(1,1) ## a random deviate to ensure that gamma[1] > gamma[2]
gamma[2] <- gamma[1] * dev
#Intercepts
alpha_mu[1] ~ dbeta(1,1)
alpha_mu[2] ~ dbeta(1,1)
#Probability of behavior switching
lambda[1] ~ dbeta(1,1)
lambda[2] <- 1 - lambda[1]
##Argos priors##
#longitudinal argos error
argos_sigma ~ dunif(0,10)
#latitidunal argos error
argos_alpha~dunif(0,10)
#correlation in argos error
rho ~ dunif(-1, 1)
}"
,fill=TRUE)sink()
## user system elapsed
## 1.516 0.033 738.5271.4 Chains
## Type Size PrettySize Rows Columns
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## m ggmap 13116336 [1] "12.5 Mb" 1280 1280
## jagM rjags.parallel 12840336 [1] "12.2 Mb" 6 NA
## b SpatialPointsDataFrame 6303360 [1] "6 Mb" 8680 47
## mxy grouped_df 2572616 [1] "2.5 Mb" 8208 52
## d SpatialPointsDataFrame 2409144 [1] "2.3 Mb" 8680 47
## sxy list 2383800 [1] "2.3 Mb" 10 NA
## oxy data.frame 2267176 [1] "2.2 Mb" 8680 47
## data list 2264824 [1] "2.2 Mb" 9 NA
## fccamlr data.frame 1649608 [1] "1.6 Mb" 41160 7## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1554325 83.1 2637877 140.9 2637877 140.9
## Vcells 9384678 71.6 28484045 217.4 44453662 339.2
Look at the convergence of phi, just for an example
Overall relationship between phi and state, nice test of convergence.
1.4.1 Compare to priors
1.5 Parameter Summary
## parameter par mean lower upper
## 1 alpha_mu alpha_mu[1] 0.72053734 0.420865193 0.97737967
## 2 alpha_mu alpha_mu[2] 0.26827412 0.016149555 0.67230532
## 3 gamma gamma[1] 0.72868602 0.618775509 0.83941352
## 4 gamma gamma[2] 0.04144941 0.002202298 0.10728724
## 5 theta theta[1] 0.01411141 -0.046330779 0.07227523
## 6 theta theta[2] 2.64299254 1.286789002 4.45539461
2 Behavioral Prediction
Relationship between phi and state
2.1 Spatial Prediction
2.2 By individual
Overlay phi and state
2.3 Compared to CMLRR regions
2.4 Autocorrelation in behavior
2.5 Location of Behavior
3 Overlap with Krill Fishery
4 Time spent in grid cell
4.1 ARS
## Type Size PrettySize Rows Columns
## mdat data.frame 16339200 [1] "15.6 Mb" 49859 47
## temp ggmap 13116288 [1] "12.5 Mb" 1280 1280
## pc tbl_df 10348896 [1] "9.9 Mb" 196300 10
## b SpatialPointsDataFrame 6303360 [1] "6 Mb" 8680 47
## mxy data.frame 2722816 [1] "2.6 Mb" 8023 59
## a tbl_df 2649704 [1] "2.5 Mb" 65900 7
## d SpatialPointsDataFrame 2409144 [1] "2.3 Mb" 8680 47
## oxy data.frame 2267176 [1] "2.2 Mb" 8680 47
## data list 2264824 [1] "2.2 Mb" 9 NA
## sxy list 1881640 [1] "1.8 Mb" 5 NA## used (Mb) gc trigger (Mb) max used (Mb)
## Ncells 1617845 86.5 2637877 140.9 2637877 140.9
## Vcells 9271797 70.8 27408683 209.2 44453662 339.2